Final answer:
The correct answer is option B. J = (-2, 15), K = -6. Using the properties of an isosceles triangle and the given gradient of HM, we found the coordinates of J and the value of K.
Step-by-step explanation:
The correct answer is option b) J = (-2, 15), K = -6.
Since triangles HJK and HMJ are isosceles with HJ=JK and triangle HMJ is right-angled at M, and given that HM has a gradient of 2, we can calculate the coordinates of M as follows:
The gradient (slope) of HM is given by (y2 - y1) / (x2 - x1), where H = (-4, 1) and M = (x, y). Since the gradient of HM is 2, (y - 1) / (x + 4) = 2.
The midpoint M of JK with J = (i,15) and K = (6,K) means it has coordinates ((i + 6)/2, (15 + K)/2). Since J and K are equidistant from H, and the x-coordinate of H is -4, the x-coordinate of J is the negative number closest to -4, which is -2. Therefore, J = (-2, 15).
Thus, using the midpoint formula, M = ((-2 + 6)/2, (15 + K)/2) = (2, (15 + K)/2). Given HM has a slope of 2, (y - 1) / (x + 4) = 2. Substituting x and y for M's coordinates gives (15 + K)/2 - 1) / (2 + 4) = 2, leading to K = -6.